All of the 5th grade teachers and students from Covington went on a field trip to an archaeology museum. Tickets were $$8.50$ each for teachers and $$3.00$ each for students, and the group paid $$41.00$ in total. The next month, the same group visited a science museum where the tickets cost $$25.50$ each for teachers and $$11.50$ each for students, and the group paid $$143.00$ in total. Find the number of teachers and students on the field trips.
Answer: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${8.5x+3y = 41}$ ${25.5x+11.5y = 143}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-3$ ${-25.5x-9y = -123}$ ${25.5x+11.5y = 143}$ Add the top and bottom equations together. $ 2.5y = 20 $ $ y = \dfrac{20}{2.5}$ ${y = 8}$ Now that you know ${y = 8}$ , plug it back into $ {8.5x+3y = 41}$ to find $x$ ${8.5x + 3}{(8)}{= 41}$ $8.5x+24 = 41$ $8.5x = 17$ $x = \dfrac{17}{8.5}$ ${x = 2}$ You can also plug ${y = 8}$ into $ {25.5x+11.5y = 143}$ and get the same answer for $x$ ${25.5x + 11.5}{(8)}{= 143}$ ${x = 2}$ There were $2$ teachers and $8$ students on the field trips.